The extended finite element method in endodontics: A scoping review and future directions for cyclic fatigue testing of nickel–titanium instruments

Abstract Objectives The present study reviews the current literature regarding the utilization of the extended finite element method (XFEM) in clinical and experimental endodontic studies and the suitability of XFEM in the assessment of cyclic fatigue in rotary endodontic nickel–titanium (NiTi) instruments. Material and Methods An electronic literature search was conducted using the appropriate search terms, and the titles and abstracts were screened for relevance. The search yielded 13 hits after duplicates were removed, and four studies met the inclusion criteria for review. Results No studies to date have utilized XFEM to study cyclic fatigue or crack propagation in rotary endodontic NiTi instruments. Challenges such as modelling material inputs and fatigue criteria could explain the lack of utilization of XFEM in the analysis of mechanical behavior in NiTi instruments. Conclusions The review showed that XFEM was seldom employed in endodontic literature. Recent work suggests potential promise in using XFEM for modelling NiTi structures.


| INTRODUCTION
Fracture of engine-driven rotary nickel-titanium (NiTi) endodontic instruments can occur via cyclic fatigue; a phenomenon by which cracks propagate due to the repeated tensile-compressive while the instrument rotates freely in a curved canal (Chien et al., 2023;Pedullà et al., 2018;Pruett et al., 1997).Instrument fracture during endodontic treatment is of concern to the practicing clinician as such events could impact the prognosis of treatment should the instrument fragment compromise chemomechanical cleansing, working length control, and root canal filling (McGuigan et al., 2013;Sjogren et al., 1990).
Numerous studies on the cyclic fatigue of rotary NiTi instruments used in endodontic literature have investigated either variations to the study design or the type of proprietary rotary instrument used (Hülsmann et al., 2019).Study designs for cyclic fatigue can be classified as static or dynamic; however, there appears to be no consensus as to the best approach for analyzing cyclic fatigue, which brings into question the clinical relevance of these experiments (Hülsmann et al., 2019).
Recently, finite element analysis (FEA) has been explored as a tool to support and validate experimental findings from benchtop cyclic fatigue testing (Chien et al., 2021).FEA is a numerical analysis where a physical model is subdivided into smaller "finite" elements to form a mesh model, which can then be investigated via computational algorithms to establish the relationships between these elements (Chien et al., 2021).Multiple studies have used FEA to visualize stresses within rotary NiTi instruments during the shaping of the root canal and to provide insight into their mechanical behavior (Arbab-Chirani et al., 2011;Chien et al., 2023;Ha et al., 2015;Kim et al., 2008Kim et al., , 2009;;Lee et al., 2011).
However, FEA is limited in the assessment of crack propagation, which is the process that ultimately leads to fracture (separation) of rotary instruments (Chien et al., 2023).Essentially, a crack is a geometric discontinuity of a material, and fracture mechanics is the study of the evolution of the discontinuity (Recho, 2012).Cracks have complicated geometries with arbitrary propagation paths, and unsurprisingly this poses a challenge in fracture mechanics where propagation occurs on curved or kinked paths within threedimensional structures (Zhuang et al., 2014).Internal defects such as material interfaces, cracks, voids, and inclusions create difficulty in the meshing process for FEA (Zhuang et al., 2014).As finite element (FE) methods require piecewise differentiable polynomial approximations, they are unsuitable for problems with discontinuous solutions (Yazid et al., 2009).In the case of modelling crack propagation, the problem involves evolving discontinuities that must be regenerated at each step (Yazid et al., 2009).Furthermore, FEA only allows cracks to propagate along the element edge and not along a natural arbitrary path (Zhuang et al., 2014).
Accordingly, improvements have since been made to FEA to overcome these limitations.The extended finite element method (XFEM) is a numerical method used to model both internal and external boundaries (e.g., holes, inclusions, and cracks) without requiring the mesh to conform to these boundaries (Yazid et al., 2009).
XFEM has a significant improvement in crack modeling with the development of a partition of unity-based enrichment method for discontinuous fields (Khoei, 2014).XFEM allows for the automatic insertion and extension of cracks inside an element, and it can model propagation from element to element (Boonrawd et al., 2022).
Naturally, these advantages could be useful in the modelling of fracture events in NiTi endodontic instruments.
This study explores the current utilization of XFEM in the endodontic literature and assesses its suitability for modelling of crack propagation in NiTi endodontic instruments.).An adapted PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) literature search strategy was employed to identify suitable papers for review (Moher et al., 2009).

| LITERATURE SEARCH STRATEGY
The language of publication was restricted to English only.The inclusion criteria required studies that used XFEM in both clinical and experimental studies in the field of endodontics where the root canal system was involved.Conference abstracts were excluded from the search, and no previous reviews on this topic were identified.

| RESULTS
The search yielded 15 hits in total between PubMed and Scopus databases, of which two were duplicates and were subsequently excluded.The 13 titles and abstracts of available English articles were screened for relevance, and all met the inclusion criteria.The 13 fulltext articles were then assessed for eligibility, and a further nine articles were excluded as XFEM was not employed in the methodology.No further studies were identified using cited reference searching or snowballing (Streeton et al., 2004).In total, four articles were selected for the current review (Figure 1).In all four selected studies, endodontically treated teeth were the subject of examination.XFEM was employed to analyze stress distribution and crack propagation in enamel (Liu et al., 2021), dentine (Boonrawd et al., 2022;Liu et al., 2021;Zhang et al. 2015Zhang et al. , 2019)), and in restorative (material) interfaces such as in glass ionomer cement and adhesive resin layers (Zhang et al., 2019).None of the studies used XFEM to analyze endodontic instruments.A summary of the findings in each study is detailed in Table 1.

| DISCUSSION
The fatigue failure process can be divided into three stages, which are: 1.The formation of cracks (crack initiation).
To date, no studies have used XFEM to investigate crack phenomena in endodontic NiTi instruments.From the four studies selected, XFEM has been utilized for assessing crack propagation, but in the context of tooth survivability.All identified studies varied in the study design used to assess the impact of access cavities, ferrules, or restorative considerations on the likelihood of crack formation in the presence of a static load that is likely to be found in normal occlusal function.However, the strengths and limitations of the reviewed studies can provide insight into the suitability of XFEM for the study of cyclic fatigue failure in NiTi instruments.
In the 2015 study of Zhang et al., the numerical predictions and experimental results were in good agreement with regard to ultimate fracture failure and the fracture locations for maxillary central incisors under static load (Zhang et al., 2015).XFEM analysis clarified susceptible areas for crack formation in endodontically treated teeth and gave possible models for fracture behavior.However, the authors noted that the cracking model could not account for fatigue-induced failures from the conventional static loading of these teeth, and followed static fracture mechanics (Zhang et al., 2015).As fractures in prosthodontic restorations occur due to cyclic loading rather than a singular acute overload, static fracture mechanics may not be an accurate representation of clinical reality (Valdivia et al., 2012;Zhang et al., 2015).The same could be said about NiTi endodontic instruments used during shaping and cleaning of root canals: instrument fracture does not always occur due to loads that exceed the elastic limit of the instruments, but in situations where the instrument is allowed to rotate freely in a canal (Pruett et al., 1997).
Future study designs should consider dynamic loads over static loads to mimic clinical reality.Previous work in other biomedical applications has modelled cyclic loading in XFEM studies, such as mandibular reconstruction plates (Wan et al., 2021) and metal fiber-reinforced polymer composites (Rashnooie et al., 2023).Such concepts could be brought into the realm of loading conditions on NiTi instruments.
Studies using conventional FEA have already been able to predict fracture locations of NiTi instruments in cyclic action to a reasonable degree of accuracy (Chien et al., 2023;Scattina et 2015).
However, without a fatigue criterion or fatigue model, the prediction F I G U R E 1 Adapted PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) flow diagram detailing the literature search strategy (Moher et al., 2009). of fatigue life would still be limited using either FEA or XFEM techniques in endodontic instruments.In other words, future research could focus on the first stage of the fatigue process (crack initiation) rather than the postmortem approach of locating the region of failure.In a previous study, the fatigue phenomenon of a mandibular reconstruction plate has already been modelled with a two-phase crack initiation (by accounting for the correct material constants and the hysteresis strain energy per cycle) and propagation process (via a crack evolution model through application of the Paris' law) (Wan et al., 2022).Other studies have also suggested various equations for crack modelling (Shahmoradi et al., 2022;Tian et al., 2019).Logically, applying similar concepts for a rotary instrument inside a canal should theoretically be no different provided the correct material constants are considered for a similar two-phase crack modelling process.Three of the four included studies assumed the materials investigated to be either brittle or quasi-brittle (Liu et al., 2021;Zhang et al. 2015Zhang et al. , 2019)).Brittle materials have extremely low plasticity and ductility, and cracks can initiate without plastic deformation, leading to brittle breakage (Huang et al., 2018).On the other hand, NiTi alloys exhibit excellent ductility (Yahia et al., 2009) and thus XFEM alone may not be suitable for modelling cracks in ductile materials, given that a significant plastic region occurs before tensile failure (Kumar & Bhardwaj, 2018).To complicate matters, NiTi alloys do not exhibit linear elastic deformation, due to the formations of intermediate transformation states such as the rhombohedrally distorted phase (R-phase) when load is applied (Chien et al., 2022).Previous work has demonstrated that these regions of plastic deformation are seen before fracture (Chien et al., 2022).
In both traditional FEA and XFEM investigations, the analyses are only as accurate as the material inputs given.In all studies reviewed, mechanical properties such as Young's modulus, Poisson ratio, tensile strength, and fracture toughness of the investigated tissues and materials (such as enamel, dentin, pulp, resin, glass ionomer cement, gutta-percha, periodontal ligament, alveolar bone) were provided using experimental data sourced from previous studies.NiTi alloys used in endodontic instruments do not have readily available experimental data for mechanical inputs, as most data on these alloys is considered proprietary information or trade secrets (Zupanc et al., 2018).Many NiTi endodontic instruments in the current market have undergone various thermomechanical treatments to enhance their performance during the chemomechanical preparation of root canals.Postmanufacture processing of NiTi alloys such as Blue and Gold treatment may not necessarily alter the core composition of NiTi, but it can alter the thresholds for martensitic and austenitic transformations (Chien et al., 2022).Given that there exist over 160 proprietary endodontic instruments in the global market (Gavini et al., 2018), it may be difficult to model accurately each instrument without access to all of their individual mechanical parameters.enrichment functions, such as those detailed by Kumar and Bhardwaj in 2018, may need to be employed to account for the ductile nature of NiTi alloys when testing (Kumar & Bhardwaj, 2018).Zhang et al. in 2022 noted that crack initiation and propagation behaviors in NiTi shape memory alloy (SMA) had yet to be reported, and they were the first to model crack propagation in NiTi SMA-containing amorphous zones and phases (Zhang et al., 2022).The cohesive model is also a potential approach which has been successfully used in previous analyses of ductile crack propagation problems in metallic materials (Li et al., 2021).Such work could be translated in the context of NiTi instrument models and their cyclic behavior in a root canal, but could still be difficult due to the complexity of the physical model, material properties and rotational movement.The limitations of this study are that there were no prior works identified that employed XFEM specifically on NiTi instruments, and consequently only related studies were available for qualitative synthesis.The suggested recommendations for future studies are therefore speculative in nature.

| CONCLUSIONS
The present review revealed that XFEM has only been employed in four studies in endodontic literature that included modelling of the root canal system.Of the studies identified, XFEM usage was focused on the analysis of crack propagation for teeth and tooth models, and no studies had yet used XFEM to analyze crack propagation in NiTi alloy endodontic instruments.Modelling cyclic fatigue of rotary NiTi instruments using XFEM could prove challenging due to the complex material properties of NiTi, and this approach may be limited for the prediction of fatigue life without a failure criterion.Nevertheless, recent work shows potential promise in using XFEM for modelling NiTi structures, particularly in understanding crack propagation.

A
literature search was conducted in November 2023 in the PubMed and Scopus database using the following search terms (extended finite element method[All fields] OR XFEM[All fields]) AND (endodontic[All fields] OR nickel-titanium[All fields] OR endodontic instrument[All fields] OR root canal[All fields] OR pulp chamber[All fields] Despite these challenges, studies continue to improve and refine XFEM techniques to provide more accurate modelling.Additional T A B L E 1 Summary of FE models and XFEM utilization in selected studies.